Demonstrators march near the White House in Washington, DC, on May 1, 2014 to protest against deportations.

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WASHINGTON—The activist immigration group United We Dream, which has long pushed the White House to stem deportations of people in the U.S. illegally, is taking aim at a new target next week: Senate Democratic leaders.

The group wants Sens. Harry Reid, the majority leader, and Michael Bennet, who heads the Senate Democrats’ campaign committee, to pressure President Barack Obama to ratchet back deportations. More than 200 of the group’s leaders will be in Washington to press the case.

Both senators supported legislation providing a path to citizenship for most of the 11 million people living in the U.S. illegally, which passed the Senate last year. But the issue is stuck in the GOP-controlled House, and United We Dream says it is targeting them because they have not taken a position on whether Mr. Obama should take administration action to stem deportations on his own.

“It’s not enough to vote on immigration reform,” said Lorella Praeli, director of advocacy for United We Dream, a group formed to support people brought to the U.S. illegally as children but now advocates for their parents and others undocumented immigrants. “Democrats are not off the hook.”

It’s a fresh sign that at least one camp of immigration activists is directing its ire squarely at Democrats, even as House Republicans continue to bottle up legislation that would provide a permanent solution. Ms. Praeli said she is not concerned that focusing on Democrats will ease pressure on Republicans to move legislation.

“I think both parties are responsible. Who’s more responsible? I don’t think that’s necessarily relevant,” she said. “Both parties have a responsibility to deliver.”

Asked about the matter, a spokesman for Sen. Bennet (D., Colo.), Adam Bozzi, said Mr. Obama should act on his own if the House fails to pass legislation this summer. “If the House continues to drag its feet we encourage the administration to pursue its efforts to prevent these innocent families from being ripped apart,” he said.

A Reid aide said Senate Democrats are still discussing their position on the issue. “There is a diversity of views and we are still having a discussion amongst ourselves on how to proceed,” he said.

To be sure, many immigration advocates continue to press House Republicans to act. On Thursday, for instance, several groups held an event aimed at pressuring House Majority Leader Eric Cantor (R., Va.) to support an overhaul.

But United We Dream is one of several groups that has shifted its focus away from House Republicans and toward the White House.

“There is uncertainty about the legislative process,” said Chris Newman of the National Day Laborer Organizing Network, which has long trained it focus on the White House. “There is certainty that the president has to fix aspects of the enforcement regime that are broken.”

Frank Sharry, who heads the pro-immigration group America’s Voice, said he sees the confrontational tactics of grassroots activists complementing the legislative push by others. “We benefit from a both/and approach, not an either/or approach,” he said.

Administration officials have suggested that Mr. Obama will make modest adjustments to the deportation system so it can be more “humane.” But activists are pushing for a bigger move: affirmative protection for millions of people in the U.S. illegally but who pose no threat. Mr. Obama has suggested he doesn’t have the legal authority to do that, but advocates say they don’t believe him, partly because he said the same thing before acting in 2012 to give safe harbor to people brought to the U.S. as children.

In addition, the White House fears that any administrative action that would undercut possible legislation by feeding GOP accusations that Mr. Obama does not enforce the law.

On Thursday, Chicago Mayor Rahm Emanuel, Mr. Obama’s first White House chief of staff, said the president is right to hold off on executive action for now, to give the House a chance to act this summer. But if lawmakers continue to stall, he predicted Mr. Obama will act on his own.

“I think using that threat that ‘if you don’t perform, I will’ is … an inspiration,” he said at a forum at the Urban Institute in Washington. “[Republicans] may not be able in the House to get past their politics. They just may not. At that point the president [says] the national interest of the United States supersedes partisan politics or legislative politics. And he will act.”

That’s not good enough for United We Dream. “Everyone says ‘let’s give Speaker [John] Boehner space.’ Our position is he’s run out of time,” Ms. Praeli said. “We’re done playing the game.”

A White House spokesman, Bobby Whithorne, said advocates should keep their focus on House Republicans. “We understand their concerns, but the only way to fix our broken immigration system and provide the permanent relief that the DREAMers deserve is through legislative action,” he said. “The president will remain focused on urging congressional action, and he will continue to encourage others to do the same.”

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